Dispersion

SYLLABUS

Pure spectrum and dispersive power, Chromatic and spherical aberration, Achromatism and its applications

What is This Chapter About?

When Isaac Newton passed a beam of sunlight through a glass prism in 1666, he observed something that changed our understanding of light forever — white light separated into a continuous band of colours: red, orange, yellow, green, blue, indigo, and violet. Newton called this spread of colours a spectrum and the phenomenon dispersion. He correctly concluded that white light is not a pure, simple form of light but a mixture of all colours, each with a different property that causes it to bend by a different amount when passing through glass.

Chapter 18 of NEB Grade 11 Physics explores dispersion quantitatively — why it happens, how to measure it, and how its unwanted effects (chromatic aberration in lenses and telescopes) can be corrected. It introduces the concept of dispersive power and the elegant design of achromatic combinations. Carrying 3 teaching hours, this chapter is compact but conceptually rich and directly relevant to the design of all high-quality optical instruments.

Key Concepts You Will Learn

Why Dispersion Occurs

Dispersion occurs because the refractive index of a material depends on the wavelength of light — a property called chromatic dispersion. For most transparent materials (glass, water, diamond), the refractive index is higher for shorter wavelengths (violet light) and lower for longer wavelengths (red light). This wavelength dependence of refractive index is described by Cauchy’s formula:

[latex]n = A + \frac{B}{\lambda^2}[/latex]

where A and B are material constants and

[latex]\lambda[/latex]

is the wavelength. Since violet light has a shorter wavelength than red, it has a higher refractive index in glass and is therefore bent more — violet deviates most and red deviates least when light passes through a prism.

Pure Spectrum

A pure spectrum is one in which each colour occupies a distinct, non-overlapping band — different wavelengths are clearly separated. A pure spectrum is obtained when the light source is a narrow slit, the prism is at minimum deviation, and a lens focuses the parallel beams of different colours onto a screen. Sunlight produces a continuous pure spectrum; a gas discharge lamp produces a line spectrum with specific colours at specific positions corresponding to specific wavelengths emitted by the atoms.

An impure spectrum is one where the colours overlap because the light beam entering the prism is too wide or the system is not properly aligned. Most everyday observations of dispersion (rainbow, prism in sunlight) produce impure spectra where colours blend into one another.

Dispersive Power

The dispersive power (

[latex]\omega[/latex]

) of a prism material quantifies how much it separates different colours relative to the average deviation it produces. It is defined as:

[latex]\omega = \frac{\delta_v – \delta_r}{\delta_y}[/latex]

where [latex]\delta_v[/latex] and [latex]\delta_r[/latex] are the deviations of violet and red light respectively, and

[latex]\delta_y[/latex]

is the deviation of yellow light (taken as the mean). Using the thin prism formula

[latex]\delta = (n-1)A[/latex]

from Chapter 16:

[latex]\omega = \frac{n_v – n_r}{n_y – 1}[/latex]

The numerator [latex](n_v – n_r)[/latex] measures the difference in refractive indices for extreme colours — it quantifies how much the material separates colours. The denominator [latex](n_y – 1)[/latex] represents the mean refracting power. A material with high dispersive power (like flint glass,

[latex]\omega \approx 0.04[/latex]

) separates colours much more than one with low dispersive power (like crown glass,

[latex]\omega \approx 0.016[/latex]

).

Chromatic Aberration

A single lens acts like a prism for off-axis rays — it refracts different wavelengths by different amounts, causing them to focus at different points along the axis. Violet light (high n) focuses closer to the lens; red light (low n) focuses further away. This defect — chromatic aberration — causes coloured fringes around images formed by simple lenses and blurs the image when observed at any single focal plane.

Chromatic aberration is a serious problem in telescopes, cameras, microscopes, and binoculars — it reduces image sharpness and produces colour fringing that degrades image quality.

Spherical Aberration

Spherical aberration occurs because rays striking the outer zones of a spherical lens are refracted more strongly than rays through the centre — they focus at a different point. This causes blurring of images even for monochromatic light. Both chromatic and spherical aberration are major challenges in optical instrument design.

Achromatism — Correcting Chromatic Aberration

An achromatic combination (or achromat) is a pair of lenses — one converging, one diverging — made from different glass types (typically crown glass and flint glass) combined to produce the same focal point for two different wavelengths (usually red and violet). The two lenses are designed so that the dispersion introduced by one is exactly cancelled by the opposite dispersion of the other, while the net refraction (focusing power) is maintained.

The condition for achromatism in a thin doublet (two lenses in contact) is:

[latex]\frac{\omega_1}{f_1} + \frac{\omega_2}{f_2} = 0[/latex]

where [latex]\omega_1[/latex] and

[latex]\omega_2[/latex]

are the dispersive powers and [latex]f_1[/latex] and [latex]f_2[/latex] are the focal lengths of the two component lenses. Since dispersive powers are positive, one focal length must be positive (converging) and the other negative (diverging). Flint glass has higher dispersive power than crown glass, so the flint glass lens (diverging) can be thinner and weaker — the combination still converges light but with greatly reduced chromatic aberration.

Achromatic lenses are used in all high-quality optical instruments — camera lenses, binoculars, telescopes, and microscope objectives all use doublet or triplet achromatic combinations to produce sharp, colour-free images.

Why This Chapter Matters for NEB Exams

Dispersion is examined consistently in NEB Grade 11 Physics. The definition of dispersive power and its formula — both in terms of deviations and refractive indices — is a standard examination question. The condition for achromatism and numerical problems involving it appear regularly. The distinction between chromatic and spherical aberration, and the distinction between pure and impure spectra, are frequent short-answer questions. The physical reason why dispersion occurs — wavelength dependence of refractive index — is a conceptual question that appears in both short-answer and extended response formats.

Important Topics to Focus On

• Why dispersion occurs — wavelength dependence of refractive index, Cauchy’s formula
• Why violet deviates most and red deviates least in a prism
• Pure spectrum — conditions for obtaining it and distinction from impure spectrum
• Dispersive power:

[latex]\omega = \frac{n_v – n_r}{n_y – 1}[/latex]

— definition and calculation Chromatic aberration — explanation and its effect on image quality Spherical aberration — explanation and comparison with chromatic aberration Achromatic combination — principle and condition:

[latex]\frac{\omega_1}{f_1} + \frac{\omega_2}{f_2} = 0[/latex]

Why crown glass and flint glass are used together Applications of achromatic lenses in optical instruments Numerical problems on dispersive power and achromatism

Connection to Other Chapters

Dispersion builds directly on Chapter 16 (Refraction through Prisms) — the thin prism formula

[latex]\delta = (n-1)A[/latex]

is the starting point for dispersive power. It connects to Chapter 17 (Lenses) — chromatic aberration is a defect of lenses explained by dispersion, and achromatic combinations are lens systems designed to overcome it. Chapter 15 (Refraction at Plane Surfaces) provides the refractive index concept central to understanding why different wavelengths refract differently. In Grade 12, the wavelength dependence of refractive index connects to Chapter 12 (Diffraction) — diffraction gratings also separate colours by wavelength, and comparing prismatic dispersion with grating dispersion is instructive. Chapter 21 (Photons) connects wavelength to photon energy through

[latex]E = hf = \frac{hc}{\lambda}[/latex]

— shorter wavelength violet photons have higher energy, which explains why glass interacts more strongly with them and gives them a higher refractive index.